Now let \((\mathcal {M}_{\mathrm {c}}, T_{\mathrm {c}}, \Omega _{\mathrm {c}})\)  be a Borchers triple with an action of \(S^1\)  or the finite cyclic group \({\mathbb {Z}}_N\) 
by inner symmetry. The argument of this Section goes parallelly for both groups.
There is a unique unitary operator \(V_{\mathrm {c}}(\kappa )\)  which implements the inner symmetry by \(S^1\) .
For a \({\mathbb {Z}}_N\) -action we take \(V_{\mathrm {c}}\)  such that \(V_{\mathrm {c}}^k\)  implements the inner symmetry for \(k\in {\mathbb {Z}}_N\) .
One can choose \(Q_{\mathrm {c}}\)  such that \(V_{\mathrm {c}}(\kappa ) = e^{i2\pi \kappa Q_{\mathrm {c}}}\)  or \(V_{\mathrm {c}}= e^{i\frac{2\pi }{N}Q_{\mathrm {c}}}\) ,
respectively. For an action of \(S^1\) , \(Q_{\mathrm {c}}\)  is unique. For \({\mathbb {Z}}_N\) ,
we remark that we can concretely choose \(Q_{\mathrm {c}}\)  as follows: for each integer \(j \in [0,N-1]\) 
we put \(\hat{V_{\mathrm {c}}}(j) := \frac{1}{N}\sum _k e^{-i\frac{2\pi jk}{N}}V_{\mathrm {c}}(k)\) . It is clear that \(\hat{V_{\mathrm {c}}}(j)\) 
is an orthogonal projection and \(\hat{V_{\mathrm {c}}}(j) \hat{V_{\mathrm {c}}}(l) = 0\)  if \(j\lnot \equiv l \mod {N}\) . In other words,
\(k\mapsto V_{\mathrm {c}}^k\)  is a representation of \({\mathbb {Z}}_N\) , whose dual group is also \({\mathbb {Z}}_N\)  and we extract
the spectral projections. Then we can take \(Q_{\mathrm {c}}:= \sum _j j \hat{V_{\mathrm {c}}}(j)\) .
By this construction, it is clear that \(Q_{\mathrm {c}}\)  commutes with the translation \(T_{\mathrm {c}}\) .
An operator \(Q_{\mathrm {c}}\)  with the properties specified above is not unique since one can amplify each spectral
component by \(N\) , but if we consider the operator \(e^{i\frac{2\pi k}{N}Q_{\mathrm {c}}\otimes Q_{\mathrm {c}}}\) ,
it does not depend on the choice of \(Q_{\mathrm {c}}\) 
(see also the remark after [1]}).
One sees that \({\rm sp}\,Q_{\mathrm {c}}\subset {\mathbb {Z}}\)  since \({\mathbb {1}} = V_{\mathrm {c}}(1) = e^{i2\pi Q_{\mathrm {c}}}\)  for an \(S^1\) -action
or \({\mathbb {1}} = V_{\mathrm {c}}^N = e^{i2\pi Q_{\mathrm {c}}}\)  for a \({\mathbb {Z}}_N\) -action (or directly by our choice).
